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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Limits

NDA 2022 · Mathematics · Limits

If f(x)=(x 2+x+|x|)/(x) then what is _ x arrow 0 f(x) equal to?

If f(x)=x2+x+∣x∣xf(x)=\frac{x 2+x+|x|}{x} then what is lim⁡x→0f(x)\lim _{x \rightarrow 0} f(x) equal to?

Options

  1. A00
  2. B11
  3. C22
  4. Dlim⁡x→0f(x)\lim _{x \rightarrow 0} f(x) does not exist

Correct answer

D. lim⁡x→0f(x)\lim _{x \rightarrow 0} f(x) does not exist

Step-by-step solution

 L.H.L. =lim⁡x→0+f(x)=lim⁡x→0−=x2+x−xx=lim⁡x→0−x=0R.H.L. =lim⁡x→0+f(x)=lim⁡x→0+x2+2xx=lim⁡x→0+x+2=2∵ L.H.L ≠ R.H.L. ∴lim⁡x→0f(x) does not exist. \begin{aligned} & \text{ L.H.L. }=\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{-}}=\frac{x^2+x-x}{x}=\lim _{x \rightarrow 0^{-}} x=0 \\ & \text{R.H.L. }=\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0^{+}} \frac{x^2+2 x}{x}=\lim _{x \rightarrow 0^{+}} x+2=2 \\ & \because \quad \text{ L.H.L } \neq \text{ R.H.L. } \\ & \therefore \quad \lim _{x \rightarrow 0} f(x) \text{ does not exist. }\end{aligned}
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